Spherical functions and harmonic analysis on free groups

Figá-Talamanca, Alessandro; Picardello, Massimo A.

doi:10.1016/0022-1236(82)90108-2

Cited by 101 publications

(147 citation statements)

References 13 publications

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“…This was partly motivated by the potential applications in Harmonic Analysis of the resulting explicit formulae (see e.g. [24] and [10]). For instance, if we denote by |t| the length of an element check) if the family (G i ) i∈I is "uniformly" unitarizable, in the following sense: there is a function F : R + → R + such that, for any i ∈ I and any u.b.…”

Section: Q2mentioning

confidence: 99%

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Are Unitarizable Groups Amenable?

Pisier

2005

Progress in Mathematics

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We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group G are similar to unitary ones, is the group amenable? In §5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prove a refinement of this result involving a generalization of the class of Herz-Schur multipliers, namely the class M d (G) which is formed of all the functions f : G → C such that there are bounded functions ξ i : G → B(H i , H i−1 ) (H i Hilbert) withWe prove that if G is a non-commutative free group, for any d ≥ 1, we haveand hence there are elements of M d (G) which are not coefficients of uniformly bounded representations. In the case d = 2, Haagerup's theorem implies that M 2 (G) = M 4 (G).

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Section: Q2mentioning

confidence: 99%

“…e.g. [24]). This means that ϕ z (t) = ϕ(|t|) where ϕ is determined inductively by: ϕ(0) = 1, ϕ(1) = z and…”

Section: Proof By Theorem 210 (I) Implies (Ii) By (22) and (23)mentioning

confidence: 99%

Are Unitarizable Groups Amenable?

Pisier

2005

Progress in Mathematics

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“…The irreducibility of the representations of the principal and of the complementary series of T was proved in [5] (see also [25], [20]). We prove that every uniformly bounded representation U z is irreducible on H; actually we consider the equivalent representation m z on H s (£l).…”

Section: Irreducibility Of U Zmentioning

confidence: 99%

Irreducibility of the analytic continuation of the principal series of a free group

Mantero

Zappa

1987

J Aust Math Soc A

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“…These may be indexed by pairs (z, w) of complex numbers, so that /z ZU) is the unique multiplicative functional h such that h(A + ) = z and h{A~) = w. They may also be naturally indexed by the group 5 -{s = (s\, s 2 , S3) e C : s^S2S3 = 1}, the two indexing methods being connected by z = --(si + s 7 + S3) and w = When A = A F , srf is isomorphic to the algebra of bi-J^-invariant functions on Sf = PGL (3, F). The multiplicative functionals for the closely related algebra of bi-K-invariant functions on 5L(3, F), where K -SL (3,6), may be found in the book [7] as a very special case. However, as PGL is not simply connected, the present situation is not quite covered by that book.…”

Section: Introduction and Notationmentioning

confidence: 99%

Harmonic analysis for groups acting on triangle buildings

Cartwright

Młotkowski

1994

J Aust Math Soc A

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Let A be a thick building of type A 2 , and let V be its set of vertices. We study a commutative algebra B4 of 'averaging' operators acting on the space of complex valued functions on T. This algebra may be identified with a space of 'biradial functions' on ~V, or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of 'type-rotating' automorphisms of A, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on d and the corresponding spherical functions. We consider the C*-algebra induced by ei on £ 2 (T), find its spectrum E, prove positive definiteness of a kernel k z for each z e E, find explicitly the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings A y arising from the groups Fy introduced in [2], this involves proving that the weak closure of si is maximal abelian in the von Neumann algebra generated by the left regular representation of Ty.

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Spherical functions and harmonic analysis on free groups

Cited by 101 publications

References 13 publications

Are Unitarizable Groups Amenable?

Are Unitarizable Groups Amenable?

Irreducibility of the analytic continuation of the principal series of a free group

Harmonic analysis for groups acting on triangle buildings

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